COMP 182 Algorithmic Thinking. Relations. Luay Nakhleh Computer Science Rice University

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1 COMP 182 Algorithmic Thinking Relations Luay Nakhleh Computer Science Rice University

2 Chapter 9, Section 1-6 Reading Material

3 When we defined the Sorting Problem, we stated that to sort the list, the elements must be totally order-able. Mathematical induction doesn t apply only to positive integers; it applies to any well-ordered set. What is an order? What is a total order? What is an order that is not total? What is a well-ordered set? To answer this question, we learn about relations and special classes of relations, including orders.

4 Relations also play an important role beyond the questions on the previous slide. For example, the relational data model for representing databases is based on the concept of a relation.

5 Binary Relations Let A and B be sets. A binary relation from A to B is a subset of A B. arb denotes (a, b) 2 R a 6 Rb denotes (a, b) /2 R

6 Binary Relations and Functions If f is a function from A to B, then f is a relation from A to B. The converse is not necessarily true.

7 Relation on a Set A relation on a set A is a relation from A to A. We have already seen such a relation: E V V.

8 Properties of Relations A relation R on a set A is called reflexive if (a, a) R for every element a A. A relation R on a set A is called symmetric if (b, a) R whenever (a, b) R, for all a,b A. A relation R on a set A such that for all a,b A, if(a, b) R and (b, a) R, then a = b is called antisymmetric. A relation R on a set A is called transitive if whenever (a, b) R and (b, c) R, then (a, c) R, for all a,b,c A.

9 Properties of Relations R 1 = {(a, b) a divides b} R 2 = {(a, b) a <b} R 3 = {(a, b) a apple b} R 4 = {(a, b) a 7 b} R 5 = {(A, B) A \ B 6= ;}

10 Combining Relations Relations are sets; therefore, set operations apply to them (e.g., you can take the union of two relations). Furthermore, if R is a relation from A to B and S is a relation from B to C, then the composite of R and S, denoted by S R, is {(a, c) a 2 A, c 2 C, 9b 2 B,((a, b) 2 R ^ (b, c) 2 S)}

11 Relations Beyond Two Sets One can define a relation on any number of sets. Let A 1,A 2,...,A n be sets.an n-ary relation on these sets is a subset of A 1 A 2 A n. The sets A 1,A 2,...,A n are called the domains of the relation, and n is called its degree. Central to database theory and implementation.

12 Representing Relations Assume A={a 1,a 2,,a m } and B={b 1,b 2,,b n }. A relation from A to B can be represented with an m n matrix M where: m ij = { 1if(ai,b j ) R, 0if(a i,b j )/ R.

13 Representing Relations Relations can also be represented using digraphs, but ones that allow self loops. a b d c

14 Closures of Relations Let R be a relation on set A. Let P be some property of relations (e.g., reflexivity). If there is a relation S with property P containing R such that S is a subset of every relation with property P containing R, then S is called the closure of R with respect to P.

15 Closures of Relations Let R be a relation on set A. Reflexive closure of R: Symmetric closure of R: S = R [{(a, a) :a 2 A} S = R [{(a, b) :(b, a) 2 R}

16 Transitive Closure Let R be a relation on set A. Recall that R can be represented as a digraph. R n is a relation on set that satisfies: (a,b) R n iff there is a path of length n from a to b in the graph of relation R. The transitive closure of R, denoted by R*, is R = n=1 R n.

17 Transitive Closure Can you devise an algorithm for computing the transitive closure of a relation? Hint: Think of matrix multiplication!

18 Equivalence Relations A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Two elements a and bthat are related by an equivalence relation are called equivalent. The notation a bis often used to denote that a and bare equivalent elements with respect to a particular equivalence relation. R = {(a, b) a m b}

19 Equivalence Relations Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a] R. When only one relation is under consideration, we can delete the subscript R and write [a] for this equivalence class. [a] R ={s (a, s) R} [ ] Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition {A i i I} of the set S, there is an equivalence relation R that has the sets A i, i I, as its equivalence classes.

20 Partial Orders

21 A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R). Members of S are called elements of the poset. S 1 = N R 1 = {(a, b) a apple b} S 2 =2 {1,2,3,4,5,6} R 2 = {(A, B) A B} The elements a and b of a poset (S, ) are called comparable if either a b or b a. When a and b are elements of S such that neither a b nor b a, a and b are called incomparable.

22 Total Order If (S, ) is a poset and every two elements of S are comparable, S is called a totally ordered or linearly ordered set, and is called a total order or a linear order. A totally ordered set is also called a chain. S 1 = N R 1 = {(a, b) a apple b} S 2 =2 {1,2,3,4,5,6} R 2 = {(A, B) A B}

23 <latexit sha1_base64="nffla+vvvyeeoxpyxay0wmamy8g=">aaacexicbvdlssnafj34rpuvdelmsagfosrf0i1qdooyon1ge8nkom2hth7m3agl9bfc+ctuxcji1p07/8zjm4vtpxdhcm693huphwuuwlj+jkxlldw19cjgcxnre2fx3ntvqiirldvojclz9oligoesarwea8eskcaxrouprzk/9cik4lf4b6oyuqhph7zhkqetewb51qvic5w6aygb7+p78cmjdoahtm2knlmyktyeejhyosmhhhxp/ha6eu0cfgivrkmobcxgpkqcp4kni06iwezokprzr9oq6jvuovloji+10sw9sookau/uvxmpczqabb7uzg5u814m/ud1euiduykp4wryskeleonaeoeshtzlkleqi00ilvzfiumasejbh1juidjzly+szrviwxx75rruu8zjkkbdditkyeznqiauur01eevp6aw9oxfj2xg1pozpaeuskc8cobkyx78hpjvk</latexit> <latexit sha1_base64="nffla+vvvyeeoxpyxay0wmamy8g=">aaacexicbvdlssnafj34rpuvdelmsagfosrf0i1qdooyon1ge8nkom2hth7m3agl9bfc+ctuxcji1p07/8zjm4vtpxdhcm693huphwuuwlj+jkxlldw19cjgcxnre2fx3ntvqiirldvojclz9oligoesarwea8eskcaxrouprzk/9cik4lf4b6oyuqhph7zhkqetewb51qvic5w6aygb7+p78cmjdoahtm2knlmyktyeejhyosmhhhxp/ha6eu0cfgivrkmobcxgpkqcp4kni06iwezokprzr9oq6jvuovloji+10sw9sookau/uvxmpczqabb7uzg5u814m/ud1euiduykp4wryskeleonaeoeshtzlkleqi00ilvzfiumasejbh1juidjzly+szrviwxx75rruu8zjkkbdditkyeznqiauur01eevp6aw9oxfj2xg1pozpaeuskc8cobkyx78hpjvk</latexit> <latexit sha1_base64="nffla+vvvyeeoxpyxay0wmamy8g=">aaacexicbvdlssnafj34rpuvdelmsagfosrf0i1qdooyon1ge8nkom2hth7m3agl9bfc+ctuxcji1p07/8zjm4vtpxdhcm693huphwuuwlj+jkxlldw19cjgcxnre2fx3ntvqiirldvojclz9oligoesarwea8eskcaxrouprzk/9cik4lf4b6oyuqhph7zhkqetewb51qvic5w6aygb7+p78cmjdoahtm2knlmyktyeejhyosmhhhxp/ha6eu0cfgivrkmobcxgpkqcp4kni06iwezokprzr9oq6jvuovloji+10sw9sookau/uvxmpczqabb7uzg5u814m/ud1euiduykp4wryskeleonaeoeshtzlkleqi00ilvzfiumasejbh1juidjzly+szrviwxx75rruu8zjkkbdditkyeznqiauur01eevp6aw9oxfj2xg1pozpaeuskc8cobkyx78hpjvk</latexit> <latexit sha1_base64="nffla+vvvyeeoxpyxay0wmamy8g=">aaacexicbvdlssnafj34rpuvdelmsagfosrf0i1qdooyon1ge8nkom2hth7m3agl9bfc+ctuxcji1p07/8zjm4vtpxdhcm693huphwuuwlj+jkxlldw19cjgcxnre2fx3ntvqiirldvojclz9oligoesarwea8eskcaxrouprzk/9cik4lf4b6oyuqhph7zhkqetewb51qvic5w6aygb7+p78cmjdoahtm2knlmyktyeejhyosmhhhxp/ha6eu0cfgivrkmobcxgpkqcp4kni06iwezokprzr9oq6jvuovloji+10sw9sookau/uvxmpczqabb7uzg5u814m/ud1euiduykp4wryskeleonaeoeshtzlkleqi00ilvzfiumasejbh1juidjzly+szrviwxx75rruu8zjkkbdditkyeznqiauur01eevp6aw9oxfj2xg1pozpaeuskc8cobkyx78hpjvk</latexit> (S, ) is a well-ordered set if it is a poset such that is a total ordering and every nonempty subset of S has a least element. S 1 = N R 1 = {(a, b) a apple b} S 2 = Z + Z + R 2 = {((a 1,b 1 ), (a 2,b 2 )) : (a 1 <a 2 ) _ (a 1 = a 2 ^ b 1 <b 2 )}

24 THE PRINCIPLE OF WELL-ORDERED INDUCTION Suppose that S is a wellordered set. Then P (x) is true for all x S, if INDUCTIVE STEP: For every y S, if P (x) is true for all x S with x y, then P (y) is true.

25 Hasse Diagrams Let (S,R) be a finite poset. The Hasse diagram of (S,R) is a digraph g=(v,e) where V=S R={(a,b) arb and c (arc and crb)}

26 Hasse Diagrams 4 GURE 2 Const 3 ({1, 2, 3, 4}, ) 2 1

27 Hasse Diagrams {a, b, c} {a,c} {a,b} {b, c} {a} {c} {b} Hasse diagram of (P({a,b,c}), )

28 Maximal and Minimal Elements Let (S,R) be a poset and a,b S. Element a is maximal if there is no element c S (c a) such that arc. Element b is minimal if there is no element c S (c b) such that crb.

29 Greatest and Least Elements Let (S,R) be a poset and a,b S. Element a is the greatest element if for every element c S, cra. Element b is the least element if for every element c S, brc. The greatest and least elements are unique when they exist.

30 What are the maximal, minimal, greatest, and least elements of the posets given by the following four Hasse diagrams? b c d d e d d c c b c a a b a b a (a) (b) (c) (d)

31 Questions?